Maximal estimates for the $(C,\alpha )$ means of $d$-dimensional Walsh-Fourier series
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Abstract:
The $d$-dimensional dyadic martingale Hardy spaces $H_{p}$ are introduced and it is proved that the maximal operator of the $(C,\alpha )$ $(\alpha =(\alpha _{1},\ldots ,\alpha _{d}))$ means of a Walsh-Fourier series is bounded from $H_{p}$ to $L_{p}$ $(1/(\alpha _{k}+1)<p<\infty )$ and is of weak type $(L_{1},L_{1})$, provided that the supremum in the maximal operator is taken over a positive cone. As a consequence we obtain that the $(C,\alpha )$ means of a function $f \in L_{1}$ converge a.e. to the function in question. Moreover, we prove that the $(C,\alpha )$ means are uniformly bounded on $H_{p}$ whenever $1/(\alpha _{k}+ 1)<p < \infty$. Thus, in case $f \in H_{p}$, the $(C,\alpha )$ means converge to $f$ in $H_{p}$ norm. The same results are proved for the conjugate $(C,\alpha )$ means, too.References
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Additional Information
- Ferenc Weisz
- Affiliation: Department of Numerical Analysis, Eötvös L. University, H-1117 Budapest, Pázmány P. sétány 1/D, Hungary
- Address at time of publication: Department of Mathematics, Humboldt University, D-10099 Berlin, Unter den Linden 6, Germany
- MR Author ID: 294049
- ORCID: 0000-0002-7766-2745
- Email: weisz@ludens.elte.hu
- Received by editor(s): September 16, 1998
- Published electronically: November 29, 1999
- Additional Notes: This research was done while the author was visiting the Humboldt University in Berlin and was supported by the Alexander von Humboldt Foundation.
- Communicated by: Christopher D. Sogge
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 2337-2345
- MSC (1991): Primary 42C10, 43A75; Secondary 60G42, 42B30
- DOI: https://doi.org/10.1090/S0002-9939-99-05368-X
- MathSciNet review: 1664379